최단시간강하곡선 문제(Brachistochrone problem) - 편집 역사

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메타데이터: 새 문단

Pythagoras0: /* 노트 */ 새 문단

2021-02-24T02:09:51Z

노트: 새 문단

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2021-02-24T02:06:05Z

메타데이터

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← 이전 판		2021년 2월 24일 (수) 02:09 판
125번째 줄:		125번째 줄:
	===소스===		===소스===
	<references />		<references />
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q529985 Q529985]
		+	===Spacy 패턴 목록===
		+	* [{'LOWER': 'brachistochrone'}, {'LEMMA': 'curve'}]
		+	* [{'LEMMA': 'brachistochrone'}]

← 이전 판		2021년 2월 24일 (수) 02:09 판
95번째 줄:		95번째 줄:
	* [http://curvebank.calstatela.edu/brach/brach.htm The Brachistochrone]		* [http://curvebank.calstatela.edu/brach/brach.htm The Brachistochrone]
	[[분류:곡선]]		[[분류:곡선]]
		+
		+	== 노트 ==
		+
		+	===말뭉치===
		+	# More specifically, the brachistochrone can use up to a complete rotation of the cycloid (at the limit when A and B are at the same level), but always starts at a cusp.<ref name="ref_9b3bb6c7">[https://en.wikipedia.org/wiki/Brachistochrone_curve Brachistochrone curve]</ref>
		+	# According to Newtonian scholar Tom Whiteside, in an attempt to outdo his brother, Jakob Bernoulli created a harder version of the brachistochrone problem.<ref name="ref_9b3bb6c7" />
		+	# Johann Bernoulli's direct method is historically important as it was the first proof that the brachistochrone is the cycloid.<ref name="ref_9b3bb6c7" />
		+	# In this blog post, we demonstrate how to use built-in mathematical expressions and the Optimization Module in COMSOL Multiphysics to solve for the brachistochrone curve.<ref name="ref_264bcfed">[https://www.comsol.com/blogs/how-to-solve-for-the-brachistochrone-curve-between-points/ How to Solve for the Brachistochrone Curve Between Points]</ref>
		+	# The brachistochrone curve is an idealized curve that provides the fastest descent possible.<ref name="ref_264bcfed" />
		+	# Next, we use an interpolation function to approximate the brachistochrone curve.<ref name="ref_264bcfed" />
		+	# The brachistochrone curve is a classic physics problem, that derives the fastest path between two points A and B which are at different elevations.<ref name="ref_aeddd8c4">[https://www.instructables.com/The-Brachistochrone-Curve/ The Brachistochrone Curve]</ref>
		+	# There is no better way to learn than through STEM, so follow on to make your very own working brachistochrone model.<ref name="ref_aeddd8c4" />
		+	# Before I end I must voice once more the admiration I feel for the unexpected identity of Huygens' tautochrone and my brachistochrone.<ref name="ref_4722e185">[https://mathshistory.st-andrews.ac.uk/HistTopics/Brachistochrone/ Brachistochrone problem]</ref>
		+	# This article presents the problem of quickest descent, or the Brachistochrone curve, that may be solved by the calculus of variations and the Euler-Lagrange equation.<ref name="ref_68ed8ad5">[https://www.tau.ac.il/~flaxer/edu/course/computerappl/exercise/Brachistochrone%20Curve.pdf Osaka keidai ronshu, vol. 61 no. 6 march 2011]</ref>
		+	# The rst step in the solution of the Euler-Lagrange equation for the brachistochrone problem: is to reduce it to a rst-order equation.<ref name="ref_ef0bb1d5">[http://www.math.utk.edu/~freire/teaching/m231f08/m231f08brachistochrone.pdf The brachistochrone problem.]</ref>
		+	# the red cycloid beats the other two It can also be asked what the brachistochrone curve among the curves joining two points and having a given shape would be.<ref name="ref_b32ffb35">[https://mathcurve.com/courbes2d.gb/brachistochrone/brachistochrone.shtml Brachistochrone]</ref>
		+	# For example, for two points at the same altitude and V-shaped curves, the brachistochrone curve is the one for which the angle of the V is a right angle, as is shown in the animation opposite.<ref name="ref_b32ffb35" />
		+	# The problem of the brachistochrone with given length is studied on this page.<ref name="ref_b32ffb35" />
		+	# One can also try to find the brachistochrone "with friction".<ref name="ref_b32ffb35" />
		+	# The brachistochrone problem was one of the earliest problems posed in the calculus of variations.<ref name="ref_f5cca441">[https://mathworld.wolfram.com/BrachistochroneProblem.html Brachistochrone Problem -- from Wolfram MathWorld]</ref>
		+	# Brachistochrone, the planar curve on which a body subjected only to the force of gravity will slide (without friction) between two points in the least possible time.<ref name="ref_5bcf547e">[https://www.britannica.com/science/brachistochrone Brachistochrone \| physics]</ref>
		+	# The name brachistochrone comes from two Greek words, brachistos meaning shortest, and chronos meaning time.<ref name="ref_02e66dfd">[https://www.myphysicslab.com/roller/brachistochrone-en.html myPhysicsLab Brachistochrone]</ref>
		+	# The brachistochrone curve can be generated by tracking a point on the rim of a wheel as it rolls on the ground.<ref name="ref_02e66dfd" />
		+	# This mathematical challenge is known as the problem of the brachistochrone.<ref name="ref_e19ac3b2">[https://medium.com/cantors-paradise/the-famous-problem-of-the-brachistochrone-8b955d24bdf7 The Famous Problem of the Brachistochrone]</ref>
		+	# Figure 3: Newton’s handwritten solution to the brachistochrone problem (source).<ref name="ref_e19ac3b2" />
		+	# The classical problem in calculus of variation is the so called brachistochrone problem 1 posed (and solved) by Bernoulli in 1696.<ref name="ref_69295678">[https://wiki.math.ntnu.no/_media/tma4180/2015v/calcvar.pdf Basics of calculus of variations]</ref>
		+	# Abstract: This article presents the problem of quickest descent, or the Brachistochrone curve, that may be solved by the calculus of variations and the Euler-Lagrange equation.<ref name="ref_b10a7db6">[https://ijpam.eu/contents/2013-82-3/8/8.pdf International journal of pure and applied mathematics]</ref>
		+	# Historically and pedagogically, the prototype problem introducing the cal- culus of variations is the brachistochrone, from the Greek for shortest time.<ref name="ref_15b1f6a4">[http://www.hep.caltech.edu/~fcp/math/variationalCalculus/variationalCalculus.pdf Physics 129a]</ref>
		+	===소스===
		+	<references />

← 이전 판		2021년 2월 24일 (수) 02:06 판
95번째 줄:		95번째 줄:
	* [http://curvebank.calstatela.edu/brach/brach.htm The Brachistochrone]		* [http://curvebank.calstatela.edu/brach/brach.htm The Brachistochrone]
	[[분류:곡선]]		[[분류:곡선]]
−
−	~~==메타데이터==~~
−	~~===위키데이터===~~
−	* ID : [https://www.wikidata.org/wiki/Q847120 Q847120]
−	~~===Spacy 패턴 목록===~~
−	* [{'LOWER': 'half'}, {'OP': '*'}, {'LEMMA': 'pipe'}]
−	* [{'LEMMA': 'halfpipe'}]

@@ 96번째 줄: / 96번째 줄: @@
 [[분류:곡선]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q847120 Q847120]

← 이전 판		2020년 12월 28일 (월) 13:17 판
95번째 줄:		95번째 줄:
	* [http://curvebank.calstatela.edu/brach/brach.htm The Brachistochrone]		* [http://curvebank.calstatela.edu/brach/brach.htm The Brachistochrone]
	[[분류:곡선]]		[[분류:곡선]]
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q847120 Q847120]

@@ 1번째 줄: / 1번째 줄: @@
 ==개요==
-* 중력을 받고 있는 물체가 정지상태에서 출발하여 가장 짧은 시간내에 하강하기 위해서 따라야 하는 곡선
+* 중력을 받고 있는 물체가 정지상태에서 출발하여 가장 짧은 시간내에 하강하기 위해서 따라야 하는 곡선
 * 1697년에 베르누이에 의하여 답이 출판
 [[파일:4402517-ParabNickF.gif]]
-−
 [http://books.google.com/books?id=dptKVr-5LJAC&pg=PA223&sig=PVA7Q1U_MyXinobyhOf54BwjShQ&hl=en#v=onepage&q&f=false Classical Mechanics]
@@ 15번째 줄: / 15번째 줄: @@
 <math>t=\int \frac{1}{v} \, ds</math>(v는 속력, ds 는 길이요소, t는 시간)
-에너지 보존 법칙 <math>mgy=\frac{1}{2}mv^2</math>  에서<math>v=\sqrt{2gy}</math>.
+에너지 보존 법칙 <math>mgy=\frac{1}{2}mv^2</math>  에서<math>v=\sqrt{2gy}</math>.
 이제 곡선의 x좌표를 y의 함수로 생각하자. 곡선을 따라 내려올 때 걸리는 시간은
@@ 22번째 줄: / 22번째 줄: @@
 문제의 정의에 따라 이 적분값을 최소가 되게 하는 곡선을 찾아야 한다.
-<math>F(y,x,x')=\frac{\sqrt{1+(x')^2}}{\sqrt{y}}</math> 에 대하여 [[오일러-라그랑지 방정식]] 을 적용하면,
+<math>F(y,x,x')=\frac{\sqrt{1+(x')^2}}{\sqrt{y}}</math> 에 대하여 [[오일러-라그랑지 방정식]] 을 적용하면,
 :<math>0 =\frac{\partial F}{\partial x} - \frac{d}{dy} \frac{\partial F}{\partial x'}=-\frac{d}{dy}(\frac{x'(y)}{\sqrt{y(1+x'(y)^2)}})</math>
@@ 38번째 줄: / 38번째 줄: @@
 따라서 사이클로이드를 얻었다.■
-−
-−
 ==재미있는 사실==
@@ 53번째 줄: / 53번째 줄: @@
 ** 최단시간강하 곡선, 최속강하선, 최단강하선
-−
 ==관련된 항목들==
@@ 59번째 줄: / 59번째 줄: @@
 * [[오일러-라그랑지 방정식]]
-−
 ==매스매티카 파일 및 계산 리소스==
 * http://demonstrations.wolfram.com/TheGreatBrachistochroneRace/
-−
-==사전 형태의 자료==
+==사전 형태의 자료==
 * http://ko.wikipedia.org/wiki/
@@ 72번째 줄: / 72번째 줄: @@
 * http://mathworld.wolfram.com/BrachistochroneProblem.html
-−
-−
 ==관련논문==
@@ 84번째 줄: / 84번째 줄: @@
-−
 ==관련도서==

@@ 78번째 줄: / 78번째 줄: @@
 ==관련논문==
-* [http://www.jstor.org/stable/4146894 The Brachistochrone Problem]<br>
+* [http://www.jstor.org/stable/4146894 The Brachistochrone Problem]
 ** Nils P. Johnson, The College Mathematics Journal, Vol. 35, No. 3 (May, 2004), pp. 192-197
-* [http://www.jstor.org/stable/2974953 Exploring the Brachistochrone Problem]<br>
+* [http://www.jstor.org/stable/2974953 Exploring the Brachistochrone Problem]
 ** LaDawn Haws, Terry Kiser, The American Mathematical Monthly, Vol. 102, No. 4 (Apr., 1995), pp. 328-336

@@ 45번째 줄: / 45번째 줄: @@
 * http://en.wikipedia.org/wiki/Half-pipe ?
 * Half-Pipe Skateboarding ?
@@ 53번째 줄: / 60번째 줄: @@
+==매스매티카 파일 및 계산 리소스==
 ==사전 형태의 자료==

@@ 27번째 줄: / 27번째 줄: @@
 적당한 상수 a에 대하여 <math>\frac{x'(y)}{\sqrt{y(1+x'(y)^2)}}=\frac{1}{\sqrt{2a}}</math>라 두자.
-이를 풀면 미분방정식  <math>\frac{dx}{dy}=\sqrt{{\frac{y}{2a-y}}</math> 를 얻는다.
+이를 풀면 다음의 미분방정식을 얻는다.
 (미분방정식의 여러 해에 대한 논의는 http://whistleralley.com/brachistochrone/brachistochrone.htm)
-<math>y=2a\sin^2\frac{\theta}{2}=a(1-\cos\theta)</math>로 치환하면, <math>x=a(\theta-\sin\theta)</math>를 얻는다.
+<math>x=\int_{0}^{y}\sqrt{\frac{y}{2a-y}}dy</math>, <math>y=2a\sin^2\frac{\theta}{2}=a(1-\cos\theta)</math>로 치환하면, <math>x=a(\theta-\sin\theta)</math>를 얻는다.
 여기서 상수 a는 주어진 점 <math>(x_1,y_1)</math>를 지날 수 있는 값으로 결정된다.